A class contains 10 men and 20 women, among which half the men and half of the women have brown-eyes. Find the probability that a person, chosen at random, is a man or has brown eyes.

Answer

To solve the problem, we need to find the probability that a person chosen at random is either a man or has brown eyes. We will use the principle of inclusion-exclusion to calculate this probability.

### Step-by-Step Solution:

1. **Determine the total number of people:**
- There are 10 men and 20 women in the class.
- Total number of people = Number of men + Number of women = 10 + 20 = **30**.

2. **Determine the number of men:**
- The number of men in the class is **10**.

3. **Determine the number of people with brown eyes:**
- Half of the men have brown eyes:
- Brown-eyed men = 10 / 2 = **5**.
- Half of the women have brown eyes:
- Brown-eyed women = 20 / 2 = **10**.
- Total number of people with brown eyes = Brown-eyed men + Brown-eyed women = 5 + 10 = **15**.

4. **Calculate the probability of selecting a man:**
- Probability of selecting a man = Number of men / Total number of people = 10 / 30 = **1/3**.

5. **Calculate the probability of selecting a person with brown eyes:**
- Probability of selecting a person with brown eyes = Number of people with brown eyes / Total number of people = 15 / 30 = **1/2**.

6. **Calculate the probability of selecting a man with brown eyes:**
- Since we already counted the 5 brown-eyed men in both previous probabilities, we need to account for this overlap.
- Probability of selecting a man with brown eyes = Number of brown-eyed men / Total number of people = 5 / 30 = **1/6**.

7. **Use the principle of inclusion-exclusion:**
- Probability (Man or Brown Eyes) = Probability (Man) + Probability (Brown Eyes) - Probability (Man and Brown Eyes)
- = (1/3) + (1/2) - (1/6).

8. **Finding a common denominator:**
- The common denominator for 3, 2, and 6 is 6.
- Convert each fraction:
- (1/3) = 2/6,
- (1/2) = 3/6,
- (1/6) = 1/6.

9. **Combine the probabilities:**
- Probability (Man or Brown Eyes) = 2/6 + 3/6 - 1/6 = (2 + 3 - 1) / 6 = 4/6 = **2/3**.

10. **Final answer:**
- The probability that a person chosen at random is a man or has brown eyes is **2/3**.